{"id":15300,"date":"2021-10-11T22:52:37","date_gmt":"2021-10-11T22:52:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/introduction-characteristics-of-functions-and-their-graphs\/"},"modified":"2021-10-25T02:13:01","modified_gmt":"2021-10-25T02:13:01","slug":"functions-and-function-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/functions-and-function-notation\/","title":{"raw":"Functions and Function Notation","rendered":"Functions and Function Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li>Define basic characteristics of functions.<\/li>\r\n \t<li>Recognize multiple representations of functions.<\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Determine whether a relation represents a function.<\/span><\/li>\r\n \t<li>Properly use a function notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\r\n\r\n<h2>Basic definitions<\/h2>\r\nA <strong>relation<\/strong> is a set of ordered pairs. The set of the first components of each <strong>ordered pair<\/strong> is called the <strong>domain <\/strong>of the relation\u00a0and the set of the second components of each ordered pair is called the <strong>range\u00a0<\/strong>of the relation. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice the first.\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right),\\left(4,8\\right),\\left(5,10\\right)\\right\\}[\/latex]<\/p>\r\nThe domain is [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0The range is [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].\r\n\r\nNote the values in the domain are also known as an <strong>input<\/strong> values, or values of the\u00a0<strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex]. Values in the range are also known as an <strong>output<\/strong> values, or values of the\u00a0<strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].\r\n\r\nA <strong>function<\/strong> [latex]f[\/latex] is a relation that assigns a single value in the range to each value in the domain<em>.<\/em> In other words, no [latex]x[\/latex]-values are used more than once. For our example that relates the first five <strong>natural numbers<\/strong> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex], is paired with exactly one element in the range, [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].\r\n\r\nNow let\u2019s consider the set of ordered pairs that relates the terms \"even\" and \"odd\" to the first five natural numbers. It would appear as\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},1\\right),\\left(\\text{even},2\\right),\\left(\\text{odd},3\\right),\\left(\\text{even},4\\right),\\left(\\text{odd},5\\right)\\right\\}[\/latex]<\/p>\r\nNotice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{odd}\\right\\}[\/latex]\u00a0is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0For example, the term \"odd\" corresponds to three values from the domain, [latex]\\left\\{1,3,5\\right\\}[\/latex]\u00a0and the term \"even\" corresponds to two values from the range, [latex]\\left\\{2,4\\right\\}[\/latex].\u00a0This violates the definition of a function, so this relation is not a function.\r\n\r\nThis image compares relations that are functions and not functions.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190946\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/> (a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[\/latex] is associated with two different outputs.[\/caption]\r\n<div class=\"textbox\">\r\n<h3>A General Note: FunctionS<\/h3>\r\nA <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say \"the output is a function of the input.\"\r\n\r\nThe <strong>input<\/strong> values make up the <strong>domain<\/strong>, and the <strong>output<\/strong> values make up the <strong>range<\/strong>.\r\n\r\n<\/div>\r\n<h2>Representing Functions Using Tables<\/h2>\r\nA common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values.\u00a0In some cases these values represent all we know about the relationship; other times the table provides a few select examples from a more complete relationship.\r\n\r\nThe table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[\/latex], where [latex]D=f\\left(m\\right)[\/latex] identifies months by an integer rather than by name.\r\n<table summary=\"Two rows and thirteen columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Month number, [latex]m[\/latex] (input)<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<td>9<\/td>\r\n<td>10<\/td>\r\n<td>11<\/td>\r\n<td>12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Days in month, [latex]D[\/latex] (output)<\/strong><\/td>\r\n<td>31<\/td>\r\n<td>28<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below\u00a0defines a function [latex]Q=g\\left(n\\right)[\/latex]. Remember, this notation tells us that [latex]g[\/latex] is the name of the function that takes the input [latex]n[\/latex] and gives the output [latex]Q[\/latex].\r\n<table summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]n[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]Q[\/latex]<\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Age in years, [latex]\\text{ }a\\text{ }[\/latex] (input)<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<td>9<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Height in inches, [latex]\\text{ }h\\text{ }[\/latex] (output)<\/strong><\/td>\r\n<td>40<\/td>\r\n<td>42<\/td>\r\n<td>44<\/td>\r\n<td>47<\/td>\r\n<td>50<\/td>\r\n<td>52<\/td>\r\n<td>54<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3><strong>How To: Given a table of input and output values, determine whether the table represents a function.\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the input and output values.<\/li>\r\n \t<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Tables that Represent Functions<\/h3>\r\nWhich table, A, B, or C, represents a function (if any)?\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table A<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table B<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\u20133<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table C<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"979211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979211\"]\r\n\r\na)\u00a0and b)\u00a0define functions. In both, each input value corresponds to exactly one output value. c)\u00a0does not define a function because the input value of 5 corresponds to two different output values.\r\n\r\nWhen a table represents a function, corresponding input and output values can also be specified using function notation.\r\n\r\nThe function represented by a)\u00a0can be represented by writing\r\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=1,f\\left(5\\right)=3,\\text{and }f\\left(8\\right)=6[\/latex]<\/p>\r\nSimilarly, the statements\u00a0[latex]g\\left(-3\\right)=5,g\\left(0\\right)=1,\\text{and }g\\left(4\\right)=5[\/latex]\u00a0represent the function in b).\r\n\r\nc)\u00a0cannot be expressed in a similar way because it does not represent a function.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1729&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining If Menu Price Lists Are Functions<\/h3>\r\nThe coffee shop menu consists of items and their prices.\r\n<ol>\r\n \t<li>Is price a function of the item?<\/li>\r\n \t<li>Is the item a function of the price?<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190949\/CNX_Precalc_Figure_01_01_0042.jpg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"487\" height=\"233\" \/>\r\n[reveal-answer q=\"507796\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507796\"]\r\n<ol>\r\n \t<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190951\/CNX_Precalc_Figure_01_01_0272.jpg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"731\" height=\"241\" \/>Each item on the menu has only one price, so the price is a function of the item.<\/li>\r\n \t<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190954\/CNX_Precalc_Figure_01_01_0282.jpg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/>Therefore, the item is a not a function of price.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining If Class Grade Rules Are Functions<\/h3>\r\nIn a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Percent Grade<\/th>\r\n<td>0\u201356<\/td>\r\n<td>57\u201361<\/td>\r\n<td>62\u201366<\/td>\r\n<td>67\u201371<\/td>\r\n<td>72\u201377<\/td>\r\n<td>78\u201386<\/td>\r\n<td>87\u201391<\/td>\r\n<td>92\u2013100<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>Grade Point Average<\/th>\r\n<td>0.0<\/td>\r\n<td>1.0<\/td>\r\n<td>1.5<\/td>\r\n<td>2.0<\/td>\r\n<td>2.5<\/td>\r\n<td>3.0<\/td>\r\n<td>3.5<\/td>\r\n<td>4.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"813427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"813427\"]\r\n\r\nFor any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.\r\n\r\nIn the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe table below\u00a0lists the five greatest baseball players of all time in order of rank.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Player<\/th>\r\n<th>Rank<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Babe Ruth<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Willie Mays<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Ty Cobb<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Walter Johnson<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Hank Aaron<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Is the rank a function of the player name?<\/li>\r\n \t<li>Is the player name a function of the rank?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"112010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"112010\"]\r\n<ol>\r\n \t<li>yes<\/li>\r\n \t<li>yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Representing Functions Using Graphs<\/h2>\r\n<p id=\"fs-id1165135435786\">Another way we can represent a function is a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\r\n<p id=\"fs-id1165137637786\">The most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f[\/latex]. The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em data-effect=\"italics\">x<\/em>-coordinate of each point is an input value and the <em data-effect=\"italics\">y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 1 tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1[\/latex]. However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<span id=\"fs-id1165137572613\" data-type=\"media\" data-alt=\"Graph of a polynomial.\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"alignleft\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010534\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" data-media-type=\"image\/jpg\" \/> <b>Figure 1<\/b>[\/caption]\r\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em data-effect=\"italics\">not<\/em> define a function because a function has only one output value for each input value.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010535\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" data-media-type=\"image\/jpg\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165137804163\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165134200185\"><strong>How To: Given a graph, use the vertical line test to determine if the graph represents a function. <\/strong><\/h3>\r\n<ol id=\"fs-id1165133277614\" data-number-style=\"arabic\">\r\n \t<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, determine that the graph does not represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137761111\" class=\"textbox shaded\">\r\n<h3>Example: Applying the Vertical Line Test<\/h3>\r\nWhich of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<span id=\"fs-id1165137786563\" data-type=\"media\" data-alt=\"Graph of a polynomial.\">\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"alignleft\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" data-media-type=\"image\/jpg\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"632895\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"632895\"]If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 3. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point.\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"259\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"259\" height=\"236\" data-media-type=\"image\/jpg\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135210137\">Does the graph in Figure 15 represent a function?<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" data-media-type=\"image\/jpg\" \/> <b>Figure 15<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"783855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783855\"]\r\n\r\nYes.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\nThe following video summarizes the basic characteristics of functions as well as their different representations mentioned so far.\r\n\r\nhttps:\/\/youtu.be\/zT69oxcMhPw\r\n<h3>Using Function Notation<\/h3>\r\nOnce we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that makes it easier to work\u00a0with functions.\r\n\r\nTo represent \"height is a function of age,\" we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h\\text{ is }f\\text{ of }a &amp;&amp;\\text{We name the function }f;\\text{ height is a function of age}. \\\\ &amp;h=f\\left(a\\right) &amp;&amp;\\text{We use parentheses to indicate the function input}\\text{. } \\\\ &amp;f\\left(a\\right) &amp;&amp;\\text{We name the function }f;\\text{ the expression is read as }\"f\\text{ of }a\". \\end{align}[\/latex]<\/p>\r\nRemember, we can use any letter to name the function; we can use the notation [latex]h\\left(a\\right)[\/latex]\u00a0 to show that [latex]h[\/latex] depends on [latex]a[\/latex]. The input value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get an output value. The parentheses indicate that age is input into the function; they do not indicate multiplication.\r\n\r\nWe can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \"first add [latex]a[\/latex]\u00a0and [latex]b[\/latex], and the result is the input for the function [latex]f[\/latex].\" We must perform the operations in this order to obtain the correct result.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Function Notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as [latex]\"y[\/latex] is a function of [latex]x.\"[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Function Notation for Days in a Month<\/h3>\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month in a non-leap year.\r\n\r\n[reveal-answer q=\"349740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"349740\"]\r\n\r\nThe number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190956\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/>\r\n\r\nFor example, [latex]f\\left(\\text{April}\\right)=30[\/latex], because April has 30 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).\r\n<h4>Analysis of the Solution<\/h4>\r\nWe must restrict the function to non-leap years. Otherwise February would have 2 outputs and this would not be a function. Also note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Interpreting Function Notation<\/h3>\r\nA function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?\r\n\r\n[reveal-answer q=\"299999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"299999\"]\r\n\r\nWhen we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]N[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2510&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Instead of a notation such as [latex]y=f\\left(x\\right)[\/latex], could we use the same symbol for the output as for the function, such as [latex]y=y\\left(x\\right)[\/latex], meaning \"<em>y<\/em> is a function of <em>x<\/em>?\"<\/strong>\r\n\r\n<em>Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f[\/latex], which is a rule or procedure, and the output [latex]y[\/latex] we get by applying [latex]f[\/latex] to a particular input [latex]x[\/latex]. This is why we usually use notation such as [latex]y=f\\left(x\\right),P=W\\left(d\\right)[\/latex], and so on.<\/em>\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<div class=\"textbox\">\r\n<ol>\r\n \t<li><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying the Vertical Line Test<\/h3>\r\nWhich of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n[reveal-answer q=\"689864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689864\"]\r\n\r\nIf any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDoes the graph below represent a function?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/>\r\n[reveal-answer q=\"783855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783855\"]\r\n\r\nYes.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137851183\">\r\n \t<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\r\n \t<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\r\n \t<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\r\n \t<li>An algebraic form of a function can be written from an equation.<\/li>\r\n \t<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137758543\" class=\"definition\">\r\n \t<dt><strong>dependent variable<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137758548\">an output variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137758552\" class=\"definition\">\r\n \t<dt><strong>domain<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137932580\" class=\"definition\">\r\n \t<dt><strong>function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134149782\" class=\"definition\">\r\n \t<dt><strong>independent variable<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134149787\">an input variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135511353\" class=\"definition\">\r\n \t<dt><strong>input<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135508564\" class=\"definition\">\r\n \t<dt><strong>output<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135508573\" class=\"definition\">\r\n \t<dt><strong>range<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135315533\" class=\"definition\">\r\n \t<dt><strong>relation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dt><strong>vertical line test<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li>Define basic characteristics of functions.<\/li>\n<li>Recognize multiple representations of functions.<\/li>\n<li class=\"li2\"><span class=\"s1\">Determine whether a relation represents a function.<\/span><\/li>\n<li>Properly use a function notation.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\n<h2>Basic definitions<\/h2>\n<p>A <strong>relation<\/strong> is a set of ordered pairs. The set of the first components of each <strong>ordered pair<\/strong> is called the <strong>domain <\/strong>of the relation\u00a0and the set of the second components of each ordered pair is called the <strong>range\u00a0<\/strong>of the relation. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice the first.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right),\\left(4,8\\right),\\left(5,10\\right)\\right\\}[\/latex]<\/p>\n<p>The domain is [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0The range is [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].<\/p>\n<p>Note the values in the domain are also known as an <strong>input<\/strong> values, or values of the\u00a0<strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex]. Values in the range are also known as an <strong>output<\/strong> values, or values of the\u00a0<strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].<\/p>\n<p>A <strong>function<\/strong> [latex]f[\/latex] is a relation that assigns a single value in the range to each value in the domain<em>.<\/em> In other words, no [latex]x[\/latex]-values are used more than once. For our example that relates the first five <strong>natural numbers<\/strong> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex], is paired with exactly one element in the range, [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].<\/p>\n<p>Now let\u2019s consider the set of ordered pairs that relates the terms &#8220;even&#8221; and &#8220;odd&#8221; to the first five natural numbers. It would appear as<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},1\\right),\\left(\\text{even},2\\right),\\left(\\text{odd},3\\right),\\left(\\text{even},4\\right),\\left(\\text{odd},5\\right)\\right\\}[\/latex]<\/p>\n<p>Notice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{odd}\\right\\}[\/latex]\u00a0is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0For example, the term &#8220;odd&#8221; corresponds to three values from the domain, [latex]\\left\\{1,3,5\\right\\}[\/latex]\u00a0and the term &#8220;even&#8221; corresponds to two values from the range, [latex]\\left\\{2,4\\right\\}[\/latex].\u00a0This violates the definition of a function, so this relation is not a function.<\/p>\n<p>This image compares relations that are functions and not functions.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190946\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p class=\"wp-caption-text\">(a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[\/latex] is associated with two different outputs.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: FunctionS<\/h3>\n<p>A <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say &#8220;the output is a function of the input.&#8221;<\/p>\n<p>The <strong>input<\/strong> values make up the <strong>domain<\/strong>, and the <strong>output<\/strong> values make up the <strong>range<\/strong>.<\/p>\n<\/div>\n<h2>Representing Functions Using Tables<\/h2>\n<p>A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values.\u00a0In some cases these values represent all we know about the relationship; other times the table provides a few select examples from a more complete relationship.<\/p>\n<p>The table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[\/latex], where [latex]D=f\\left(m\\right)[\/latex] identifies months by an integer rather than by name.<\/p>\n<table summary=\"Two rows and thirteen columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Month number, [latex]m[\/latex] (input)<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<td>11<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td><strong>Days in month, [latex]D[\/latex] (output)<\/strong><\/td>\n<td>31<\/td>\n<td>28<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below\u00a0defines a function [latex]Q=g\\left(n\\right)[\/latex]. Remember, this notation tells us that [latex]g[\/latex] is the name of the function that takes the input [latex]n[\/latex] and gives the output [latex]Q[\/latex].<\/p>\n<table summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td>[latex]n[\/latex]<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>[latex]Q[\/latex]<\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Age in years, [latex]\\text{ }a\\text{ }[\/latex] (input)<\/strong><\/td>\n<td>5<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td><strong>Height in inches, [latex]\\text{ }h\\text{ }[\/latex] (output)<\/strong><\/td>\n<td>40<\/td>\n<td>42<\/td>\n<td>44<\/td>\n<td>47<\/td>\n<td>50<\/td>\n<td>52<\/td>\n<td>54<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3><strong>How To: Given a table of input and output values, determine whether the table represents a function.<br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the input and output values.<\/li>\n<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Tables that Represent Functions<\/h3>\n<p>Which table, A, B, or C, represents a function (if any)?<\/p>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table A<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table B<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u20133<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table C<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q979211\">Show Solution<\/span><\/p>\n<div id=\"q979211\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)\u00a0and b)\u00a0define functions. In both, each input value corresponds to exactly one output value. c)\u00a0does not define a function because the input value of 5 corresponds to two different output values.<\/p>\n<p>When a table represents a function, corresponding input and output values can also be specified using function notation.<\/p>\n<p>The function represented by a)\u00a0can be represented by writing<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=1,f\\left(5\\right)=3,\\text{and }f\\left(8\\right)=6[\/latex]<\/p>\n<p>Similarly, the statements\u00a0[latex]g\\left(-3\\right)=5,g\\left(0\\right)=1,\\text{and }g\\left(4\\right)=5[\/latex]\u00a0represent the function in b).<\/p>\n<p>c)\u00a0cannot be expressed in a similar way because it does not represent a function.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1729&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining If Menu Price Lists Are Functions<\/h3>\n<p>The coffee shop menu consists of items and their prices.<\/p>\n<ol>\n<li>Is price a function of the item?<\/li>\n<li>Is the item a function of the price?<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190949\/CNX_Precalc_Figure_01_01_0042.jpg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"487\" height=\"233\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q507796\">Show Solution<\/span><\/p>\n<div id=\"q507796\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190951\/CNX_Precalc_Figure_01_01_0272.jpg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.\" width=\"731\" height=\"241\" \/>Each item on the menu has only one price, so the price is a function of the item.<\/li>\n<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190954\/CNX_Precalc_Figure_01_01_0282.jpg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/>Therefore, the item is a not a function of price.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining If Class Grade Rules Are Functions<\/h3>\n<p>In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Percent Grade<\/th>\n<td>0\u201356<\/td>\n<td>57\u201361<\/td>\n<td>62\u201366<\/td>\n<td>67\u201371<\/td>\n<td>72\u201377<\/td>\n<td>78\u201386<\/td>\n<td>87\u201391<\/td>\n<td>92\u2013100<\/td>\n<\/tr>\n<tr>\n<th>Grade Point Average<\/th>\n<td>0.0<\/td>\n<td>1.0<\/td>\n<td>1.5<\/td>\n<td>2.0<\/td>\n<td>2.5<\/td>\n<td>3.0<\/td>\n<td>3.5<\/td>\n<td>4.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q813427\">Show Solution<\/span><\/p>\n<div id=\"q813427\" class=\"hidden-answer\" style=\"display: none\">\n<p>For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.<\/p>\n<p>In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The table below\u00a0lists the five greatest baseball players of all time in order of rank.<\/p>\n<table>\n<thead>\n<tr>\n<th>Player<\/th>\n<th>Rank<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Babe Ruth<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>Willie Mays<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>Ty Cobb<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>Walter Johnson<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>Hank Aaron<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Is the rank a function of the player name?<\/li>\n<li>Is the player name a function of the rank?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q112010\">Show Solution<\/span><\/p>\n<div id=\"q112010\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>yes<\/li>\n<li>yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Representing Functions Using Graphs<\/h2>\n<p id=\"fs-id1165135435786\">Another way we can represent a function is a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n<p id=\"fs-id1165137637786\">The most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f[\/latex]. The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em data-effect=\"italics\">x<\/em>-coordinate of each point is an input value and the <em data-effect=\"italics\">y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 1 tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1[\/latex]. However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<span id=\"fs-id1165137572613\" data-type=\"media\" data-alt=\"Graph of a polynomial.\"><br \/>\n<\/span><\/p>\n<div style=\"width: 741px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010534\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em data-effect=\"italics\">not<\/em> define a function because a function has only one output value for each input value.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010535\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165137804163\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165134200185\"><strong>How To: Given a graph, use the vertical line test to determine if the graph represents a function. <\/strong><\/h3>\n<ol id=\"fs-id1165133277614\" data-number-style=\"arabic\">\n<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the graph does not represent a function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137761111\" class=\"textbox shaded\">\n<h3>Example: Applying the Vertical Line Test<\/h3>\n<p>Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<span id=\"fs-id1165137786563\" data-type=\"media\" data-alt=\"Graph of a polynomial.\"><br \/>\n<\/span><\/p>\n<div style=\"width: 985px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632895\">Solution<\/span><\/p>\n<div id=\"q632895\" class=\"hidden-answer\" style=\"display: none\">If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 3. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point.<\/p>\n<div style=\"width: 269px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"259\" height=\"236\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135210137\">Does the graph in Figure 15 represent a function?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783855\">Show Solution<\/span><\/p>\n<div id=\"q783855\" class=\"hidden-answer\" style=\"display: none\">\n<p>Yes.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h3><\/h3>\n<p>The following video summarizes the basic characteristics of functions as well as their different representations mentioned so far.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determine if a Relation is a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zT69oxcMhPw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Using Function Notation<\/h3>\n<p>Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that makes it easier to work\u00a0with functions.<\/p>\n<p>To represent &#8220;height is a function of age,&#8221; we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&h\\text{ is }f\\text{ of }a &&\\text{We name the function }f;\\text{ height is a function of age}. \\\\ &h=f\\left(a\\right) &&\\text{We use parentheses to indicate the function input}\\text{. } \\\\ &f\\left(a\\right) &&\\text{We name the function }f;\\text{ the expression is read as }\"f\\text{ of }a\". \\end{align}[\/latex]<\/p>\n<p>Remember, we can use any letter to name the function; we can use the notation [latex]h\\left(a\\right)[\/latex]\u00a0 to show that [latex]h[\/latex] depends on [latex]a[\/latex]. The input value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get an output value. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n<p>We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means &#8220;first add [latex]a[\/latex]\u00a0and [latex]b[\/latex], and the result is the input for the function [latex]f[\/latex].&#8221; We must perform the operations in this order to obtain the correct result.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Function Notation<\/h3>\n<p>The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as [latex]\"y[\/latex] is a function of [latex]x.\"[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Function Notation for Days in a Month<\/h3>\n<p>Use function notation to represent a function whose input is the name of a month and output is the number of days in that month in a non-leap year.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q349740\">Show Solution<\/span><\/p>\n<div id=\"q349740\" class=\"hidden-answer\" style=\"display: none\">\n<p>The number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a &#8220;rule&#8221; that associates a specific number (the output) with each input.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190956\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/><\/p>\n<p>For example, [latex]f\\left(\\text{April}\\right)=30[\/latex], because April has 30 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We must restrict the function to non-leap years. Otherwise February would have 2 outputs and this would not be a function. Also note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Interpreting Function Notation<\/h3>\n<p>A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q299999\">Show Solution<\/span><\/p>\n<div id=\"q299999\" class=\"hidden-answer\" style=\"display: none\">\n<p>When we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]N[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2510&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Instead of a notation such as [latex]y=f\\left(x\\right)[\/latex], could we use the same symbol for the output as for the function, such as [latex]y=y\\left(x\\right)[\/latex], meaning &#8220;<em>y<\/em> is a function of <em>x<\/em>?&#8221;<\/strong><\/p>\n<p><em>Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f[\/latex], which is a rule or procedure, and the output [latex]y[\/latex] we get by applying [latex]f[\/latex] to a particular input [latex]x[\/latex]. This is why we usually use notation such as [latex]y=f\\left(x\\right),P=W\\left(d\\right)[\/latex], and so on.<\/em><\/p>\n<\/div>\n<h2><\/h2>\n<div class=\"textbox\">\n<ol>\n<li><\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying the Vertical Line Test<\/h3>\n<p>Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689864\">Show Solution<\/span><\/p>\n<div id=\"q689864\" class=\"hidden-answer\" style=\"display: none\">\n<p>If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Does the graph below represent a function?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783855\">Show Solution<\/span><\/p>\n<div id=\"q783855\" class=\"hidden-answer\" style=\"display: none\">\n<p>Yes.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137851183\">\n<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\n<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\n<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\n<li>An algebraic form of a function can be written from an equation.<\/li>\n<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137758543\" class=\"definition\">\n<dt><strong>dependent variable<\/strong><\/dt>\n<dd id=\"fs-id1165137758548\">an output variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137758552\" class=\"definition\">\n<dt><strong>domain<\/strong><\/dt>\n<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932580\" class=\"definition\">\n<dt><strong>function<\/strong><\/dt>\n<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134149782\" class=\"definition\">\n<dt><strong>independent variable<\/strong><\/dt>\n<dd id=\"fs-id1165134149787\">an input variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511353\" class=\"definition\">\n<dt><strong>input<\/strong><\/dt>\n<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508564\" class=\"definition\">\n<dt><strong>output<\/strong><\/dt>\n<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508573\" class=\"definition\">\n<dt><strong>range<\/strong><\/dt>\n<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315533\" class=\"definition\">\n<dt><strong>relation<\/strong><\/dt>\n<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dt><strong>vertical line test<\/strong><\/dt>\n<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\n<\/dl>\n<\/div>\n<h2><\/h2>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15300\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 111625, 111715, 11722. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 111699. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Function Notation Application. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\">https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Function Notation Application. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\">https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 2510, 1729. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15800. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1647. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 97486. <strong>Authored by<\/strong>: Carmichael, Patrick. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15766, 2886, 3751. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 40676. <strong>Authored by<\/strong>: Micheal Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Determine if a Relation is a Function. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zT69oxcMhPw\">https:\/\/youtu.be\/zT69oxcMhPw<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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